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Math Help - Help with very simple exponent question

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    Help with very simple exponent question

    2 ^ 2 = 4. Understood. 2 multiplied by itself 2 times. Simple.

    2 ^ 2 ^ 2 = 16. Understood. 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. Simple.

    2 ^ 2 ^ 2 ^ 2 = 65536. What? 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. 16 multiplied by itself twice, to equal 256, right? No, 65,536. Why?

    Please explain this very simply to me like I'm five. As simple as they are, exponents confuse me when they go beyond x ^ y. According to the power rule, you're supposed to multiple the exponents together right? Well, it doesn't work here from what I can tell.
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    Re: Help with very simple exponent question

    Quote Originally Posted by Nubris View Post
    2 ^ 2 = 4. Understood. 2 multiplied by itself 2 times. Simple.

    2 ^ 2 ^ 2 = 16. Understood. 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. Simple.

    2 ^ 2 ^ 2 ^ 2 = 65536. What? 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. 16 multiplied by itself twice, to equal 256, right? No, 65,536. Why?

    Please explain this very simply to me like I'm five. As simple as they are, exponents confuse me when they go beyond x ^ y. According to the power rule, you're supposed to multiple the exponents together right? Well, it doesn't work here from what I can tell.
    2^2 = 4

    2^{(2^2)} = 2^4 = 16

    2^{(2^{(2^2)})} = 2^{16} = 2^8 \cdot 2^8 = 256^2 = 65536

    -Dan
    Last edited by topsquark; September 8th 2013 at 10:09 AM.
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    Re: Help with very simple exponent question

    What mistake am I making in order to get 256? Why does it seem to flow normally up until 2 ^ 2 ^ 2 ^ 2? Before that it works using whatever mistake I'm making. Sorry, but I'm really stupid so I need this explained condescendingly.
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    Re: Help with very simple exponent question

    Quote Originally Posted by Nubris View Post
    2 ^ 2 = 4. Understood. 2 multiplied by itself 2 times. Simple.

    2 ^ 2 ^ 2 = 16. Understood. 2 multiplied by itself 2 times, equaling 4, and multiplying that by itself 2 times to equal 16. Simple.
    First we need to decide whether 2^{2^2} means \left(2^2\right)^2 or 2^{\left(2^2\right)}. It is important because, unlike addition and multiplication, exponentiation is not associative, which means that in general \left(x^y\right)^z\ne x^{\left(y^z\right)}, or, written in plain text, (x ^ y) ^ z ≠ x ^ (y ^ z). By convention, 2^{2^2} means 2^{\left(2^2\right)}, i.e., exponentiation is right-associative. In contrast, your description corresponds to \left(2^2\right)^2. Indeed, "2 multiplied by itself 2 times, equaling 4" evaluates the bottom part in parentheses, so the expression reduces to 4^2, "and multiplying that by itself 2 times to equal 16". The correct way to evaluate 2^{2^2} is to compute the top 2^2 and then raise the bottom 2 to that power. It turns out that by accident both ways of computing 2^{2^2} give the same result since 2^4 = 4^2, but this is not true for your next example. If it were \left(\left(2^2\right)^2\right)^2, then the answer indeed would be 256.
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    Forum Admin topsquark's Avatar
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    Re: Help with very simple exponent question

    2^{(2^{(2^2)})} = 2^{(2^{4})} = 2^{16}

    Was this the problem or how to get 2^{16} = 65536?

    -Dan
    Last edited by topsquark; September 9th 2013 at 06:15 PM.
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